4.6 Circle properties
- Syllabus
- 2017
- Topic
- 4.6
- Level
- Foundation
Learning objectives
4.6A Circle terminologyRecognise the centre, radius, chord, diameter, circumference, tangent, arc, sector and segment of a circle.4.6B Chord and tangent properties of circles Two tangentsunderstand chord and tangent properties of circles Two tangents from a point to a circle are equal in length Tangents are perpendicular to the radius at the point of contact The line from the centre of a circle which is perpendicular to a chord bisects the chord (and the converse)
Recognise circle terminology
A circle is the set of points at a fixed distance from its centre. That fixed distance is the radius; a diameter is a chord through the centre and has length twice the radius.
| Term | Meaning |
|---|---|
| circumference | the circle's boundary |
| chord | straight segment joining two points on the circle |
| tangent | line touching the circle at one point |
| arc | part of the circumference |
| sector | region between two radii and an arc |
| segment | region between a chord and its arc |
A diameter is always a chord, but a chord is a diameter only when it passes through the centre.
A sector has two straight radius edges; a segment has one straight chord edge. Do not name either region from appearance alone.
Use chord and tangent properties
A tangent is perpendicular to the radius at the point of contact. Two tangents drawn from the same external point have equal lengths.
| Condition | Consequence |
|---|---|
| centre-to-chord line is perpendicular | it bisects the chord |
| centre-to-chord line bisects the chord | it is perpendicular to the chord |
| equal chords | they are equally distant from the centre |
Add the radius to a tangent diagram to create a right angle. Join the centre to a chord midpoint to create two congruent right triangles when useful.
The right angle is between the tangent and the radius at the contact point, not between a tangent and every chord through that point.
Use internal and external intersecting chord properties
For chords AB and CD intersecting inside a circle at X, AXimesXB=CXimesXD.
From an external point P, if two secants meet the circle at A,B and C,D, then PAimesPB=PCimesPD, using each external length times its whole secant length.
| Step | Action |
|---|---|
| 1 | identify the common intersection point |
| 2 | label the two parts of each chord or the external and whole secant lengths |
| 3 | equate the two products |
| 4 | solve and reject impossible negative lengths |
For an external secant, the second factor is the whole length from the external point to the far circle intersection, not just the portion inside the circle.
Recognise cyclic quadrilaterals
A cyclic quadrilateral has all four vertices on one circle. The circle is its circumcircle.
| Evidence | Conclusion |
|---|---|
| four vertices lie on one circle | cyclic |
| a pair of opposite angles sums to 180∘ | cyclic |
| an exterior angle equals the opposite interior angle | cyclic |
The sides of a cyclic quadrilateral are chords of the circle; its diagonals are also chords.
A quadrilateral drawn inside a circle is not necessarily cyclic: every vertex must lie on the circumference, not merely inside the disk.
Use circle angle theorems
Angles subtended by the same chord at the circumference are equal. The angle at the centre is twice the angle at the circumference standing on the same arc, and an angle in a semicircle is 90∘.
| Configuration | Result |
|---|---|
| cyclic quadrilateral | opposite angles sum to 180∘ |
| tangent and chord | angle between them equals the angle in the alternate segment |
| two radii | they form an isosceles triangle |
If chord AC subtends 38∘ at point B, then angle AOC=76∘. Since OA=OC, each base angle in triangle AOC is (180−76)/2=52∘.
Mark the chord or arc each angle stands on before selecting a theorem, then combine with triangle, straight-line or point-angle facts.
The centre angle is double only when both angles subtend the same arc. Do not double merely because one angle is drawn near the centre.